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arXiv:2609.28213

Local time-reversal-invariant topological markers in two dimensions

Thomas Klein Kvorning, Miguel F. Martínez, Julia D. Hannukainen

Published Sep 23, 2026• 0 citations
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Abstract

Local topological markers provide a real-space characterization of topological phases and are suited to noncrystalline models. We derive a local topological marker, the Chern-Simons descendant marker, for characterizing the topology of two-dimensional time-reversal-invariant near-Gaussian states from the one-particle density matrix \(ρ\). We construct an auxiliary three-dimensional state characterized by the local Chern-Simons marker, and use dimensional reduction to obtain the Chern-Simons descendant marker for the original two-dimensional state. This derivation introduces a local Hermitian involution \(S\) that is odd under time reversal. When \([S,ρ]=0\), the local Chern-Simons descendant marker reduces to half the difference of the local Chern markers in the \(S=\pm1\) sectors. Constructing an \(S\) that commutes exactly with \(ρ\) generally requires fine-tuning, but this is not needed---the local Chern-Simon marker remains well defined whenever \(Γ=\{ρ,S\}-S\) has a spectral gap around zero, even when \([S,ρ]\neq0\). This leaves the practical problem of identifying a suitable \(S\) for a given state, which, importantly, is straightforward even when spin is not a candidate. We show this by considering both strong Rashba and Dresselhaus spin-orbit coupling, for which knowledge of the physical origin of the state reduces the general problem of choosing $S$ to a one-parameter family \(S(θ)\). We evaluate the optimal angle $θ$ maximizing the spectral gap of $Γ$ analytically assuming translation invariance, and find that the gap of $Γ$ remains open for a broad range of angles around the optimum. The same value of \(θ\) remains valid in disordered systems, providing an easy way to obtain choice of \(S\) for evaluating the local Chern-Simons descendant marker of the state.

DOI
10.48550/arxiv.2609.28213

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Local topological markers, topological invariants evaluated by local expectation values, are valuable for characterizing topological phases in materials lacking translation invariance. The Chern marker-the Chern number expressed in terms of the Fourier transformed Chern character-is an easily applicable local marker in even dimensions, but there are no analogous expressions for odd dimensions. We provide general analytic expressions for local markers for free-fermion topological states in odd dimensions protected by local symmetries: a Chiral marker, a local Z marker which in case of translation invariance is equivalent to the chiral winding number, and a Chern-Simons marker, a local Z_{2} marker characterizing all nonchiral phases in odd dimensions. We achieve this by introducing a one-parameter family P_{ϑ} of single-particle density matrices interpolating between a trivial state and the state of interest. By interpreting the parameter ϑ as an additional dimension, we calculate the Chern marker for the family P_{ϑ}. We demonstrate the practical use of these markers by characterizing the topological phases of two amorphous Hamiltonians in three dimensions: a topological superconductor (Z classification) and a topological insulator (Z_{2} classification).
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Topological states of matter were first introduced for noninteracting fermions on an infinite uniform lattice. Since then, substantial effort has been made to generalize these concepts to more complex settings. Recently, local markers have been developed that can describe the topological state of systems without translational symmetry and a well-defined gap. However, no local marker for interacting matter has been proposed yet that is capable of directly addressing an interacting system. Here, we suggest such a many-body local marker based on the single-particle Green's function. Using this marker we identify topological transitions in finite lattices of a Chern insulator with Anderson disorder and Hubbard interactions. Importantly, our proposal can be generalized to nonequilibrium systems.
We compare three complementary diagnostics for mixed Gaussian states at nonzero temperature, focusing on the Su-Schrieffer-Heeger (SSH) chain and its inversion-symmetric extension. While the ensemble geometric phase, a mixed-state generalization of the Zak phase, remains well defined at nonzero temperature, the modulus of the corresponding expectation value vanishes in the thermodynamic limit, limiting its practical use. To develop diagnostics suitable for large systems, we introduce local twist operators acting on neighboring sites, whose expectation values provide local indicators of the underlying topological phase. The topological phase is identified from the relative magnitude of these expectation values, which only requires measuring two local expectation values at nonzero temperature, together with one additional nonlocal expectation value when next-nearest-neighbor hopping is included. In addition, we generalize the local chiral marker to mixed Gaussian states, fully determined by its single-particle correlation matrix, with a nonzero purity gap in their effective single-particle Hamiltonian. The presence of a purity gap ensures that the correlation matrix can be flattened to an effective projector. Evaluating the chiral marker with respect to the band-flattened correlation matrix yields a real-space topological invariant that coincides with the winding number in the zero-temperature limit. The ensemble geometric phase, the local twist operators, and the local chiral marker provide complementary methods to characterize topology in the SSH chain beyond pure states.